EITN90 Radar and Remote Sensing Lecture 2: The Radar Range Equation

EITN90 Radar and Remote Sensing Lecture 2: The Radar Range Equation Daniel Sjöberg Department of Electrical and Information Technology Spring 2018

Outline 1 Radar Range Equation Received power Signal to noise ratio Losses Multiple pulses Application oriented RRE:s 2 Radar Search and Detection Search mode fundamentals Detection fundamentals 3 Conclusions 2 / 58

Learning outcomes of this lecture In this lecture we will Develop a physical model for the received power of a radar from a target at a distance Interpret the result in user terms and designer terms for different applications Investigate the requirements and methods of search and detection 3 / 58

Equal radiation in all directions. Isotropic radiation pattern P t = transmitted power [W] R = distance from source [m] Q i = power density [m 2 ] 6 / 58

Directional radiation pattern Stronger radiation in some directions. Transmitting antenna gain G t (θ, φ) def = lim R Q i (R,θ,φ) P t/(4πr 2 ) = 4πAe λ 2. Effective area A e and gain G t represent the same physical concept, just a scaling by 4π/λ 2. 7 / 58

Scattered power from a target Target is hit by power density Q i, and scatters the power. Radar cross section (RCS) σ def 4πR = lim 2 Q r R Q i (Q i held constant). 8 / 58

Received power Received power is P r = A e Q r, effective area A e def = λ2 4π G r. Putting everything together implies the Radar Range Equation P r = P tg t G r λ 2 σ (4π) 3 R 4 9 / 58

Radar Range Equation The radar range equation is the fundamental model for estimating the received power in a given scenario. P r = P tg t G r λ 2 σ (4π) 3 R 4 P t = peak transmitted power [W] G t = gain of transmit antenna (unitless) G r = gain of receive antenna (unitless) λ = carrier wavelength [m] σ = mean RCS of target [m 2 ] R = range from radar to target [m] 10 / 58

Radar Range Equation, db scale The decibel (db) scale is defined as P r [db] def = 10 log 10 (P r ) The logarithm function has the properties log 10 (ab) = log 10 (a) + log 10 (b), log 10 (a/b) = log 10 (a) log 10 (b), and log 10 (a b ) = b log 10 (a). The RRE is then P r [db] = P t [db] + G t [db] + G r [db] + 2 λ [db] + σ [db] 30 log 10 (4π) 4 R [db] }{{} = 33 db For quantities with physical units, it is common to introduce a reference level: ( 10 log Pr ) def 10 = P r [dbw] (W for Watt) 10 log 10 ( λ 1 m 1 W ) def = λ [dbm] (m for meter) ( 10 log σ ) def 10 = σ [dbsm] (sm for square meters) 1 m 2 11 / 58

Bistatic scenario In a bistatic scenario, with two antennas separated in space, the transmit and receive distances R t and R r are usually different: P r = P tg t G r λ 2 σ (4π) 3 R 2 t R2 r R t R r We will focus on the monostatic scenario. 12 / 58

Thermal noise The power of the thermal noise in the radar receiver is P n = kt s B = kt 0 F B where the different factors are k is Boltzmann s constant (1.38 10 23 Ws/K) T 0 is the standard temperature (290 K) T s is the system noise temperature (T s = T 0 F ) B is the instantaneous receiver bandwidth in Hz F is the noise figure of the receiver subsystem (unitless) 14 / 58

SNR version of RRE The thermal noise of the receiver can be combined with the RRE to yield the signal to noise ratio SNR = P r P n = P tg t G r λ 2 σ (4π) 3 R 4 kt 0 F B 15 / 58

SNR version of RRE The thermal noise of the receiver can be combined with the RRE to yield the signal to noise ratio SNR = P r P n = P tg t G r λ 2 σ (4π) 3 R 4 kt 0 F B The final radar performance is determined by the signal to interference ratio, where SIR = S N + C + J = P tg t G r λ 2 σ 1 (4π) 3 R 4 kt 0 F B + C + J S = signal power N = noise power C = clutter power J = jammer power Often only one of S/N, S/C or S/J is dominating. 15 / 58

Clutter The radar signal can be scattered against many other things in the background. These interfering signals are called clutter. Since the clutter scatterers are typically located close to the scatterer we want to detect, all terms in the radar equation cancel and the target signal to clutter ratio is SCR = σ σ c The clutter RCS σ c can be significant, depending on how much is being illuminated by the radar. There are two typical kinds of clutter: Surface clutter Volume clutter More on clutter in Chapter 5. 16 / 58

Surface clutter σ cs = A c σ 0 σ cs is the surface clutter radar cross section (square meters) A c is the area of the illuminated (ground or sea surface) clutter cell (square meters) σ 0 is the surface backscatter coefficient (average reflectivity per unit area) (square meters per square meters, or unitless) 17 / 58

Volume clutter σ cv = V c η σ cv is the volume clutter radar cross section (square meters) V c is the volume of the illuminated clutter cell (cubic meters) η is the volumetric backscatter coefficient (average reflectivity per unit volume) (square meters per cubic meters, or reciprocal meters) 18 / 58

Jamming Jamming is a method of disabling a radar system by sending a strong interfering signal, saturating the receiver. The received power from this signal is calculated by the one-way equation P rj = P jg j G rj λ 2 (4π) 2 R 2 jr L s P rj received power from the jammer P j transmitted power from the jammer G j gain of the jammer antenna G rj gain of the receive antenna (in direction of jammer) R jr distance between jammer and receiver L s system losses 19 / 58

Losses We have neglected a number of real-life losses in the RRE so far. The typical system loss would be the combination of several: L s = L t L a L 1r L sp where the different factors are L s is the system loss L t is the transmit loss L a is the atmospheric loss L sp is the signal processing loss with the resulting system-loss SNR (an additional factor n p can account for multiple pulses signal processing gain, see later slides) SNR = P t G t G r λ 2 σ (4π) 3 R 4 kt 0 F BL s The various factors are discussed in the following slides. 21 / 58

Transmit loss Typically waveguides, cables, circulator, directional coupler, and switch add losses on the order of L t 3 4 db. The antenna gain G may include some losses, depending on definition. Always consult datasheets! 22 / 58

Atmospheric loss Atmospheric losses depend on frequency, weather conditions, altitude, etc. Typically measured in db/km, and limits the range of the radar. 23 / 58

Atmospheric loss Typical atmospheric losses as function of frequency, at two different altitudes. Note the peaks corresponding to resonant interaction with atmosphere molecules. Further losses are due to rain, fog etc. Long range radar systems tend to operate in frequency regions with low loss, but short-range systems may use losses for isolation. 24 / 58

Receive loss Similar to transmit losses: waveguides, cables, circulator, switch, filters etc. Include losses up to the point where the noise figure F is specified. 25 / 58

Signal processing loss Even though the signal processor usually provides gain (typically on the order of n p ), the imperfections also provide some loss. 26 / 58

Signal processing loss: beam scanning Loss due to the target not being intercepted by the maximum gain of the beam. While tracking, beam can be kept on target. 27 / 58

Signal processing loss: straddle loss Discretization of range and Doppler frequencies in different processing bins may introduce loss around 1 db in both range and Doppler. The dips can be reduced by increasing bin overlap (oversampling). 28 / 58

Multiple pulses The SNR can be improved by using data from several pulses. The signal processing gain from this can be estimated as (assuming white noise) Coherent processing (both phase and amplitude): SNR(n p ) = n p SNR(1) Noncoherent processing (only amplitude): SNR(n p ) n p SNR(1) Typically, the processing gain by using multiple pulses can be estimated as np SNR(1) SNR(n p ) n p SNR(1) Using many pulses increases the measurement time. 30 / 58

Average power, coherent processing We use n p pulses, each with duration τ and repeated at Pulse Repetition Frequency (PRF=1/PRI, Pulse Repetition Interval). t τ PRI T d = n p PRI Dwell time T d = n p PRI = n p /PRF. Duty cycle d t = τ/pri = τ PRF, τ = 1/B. The coherent processing SNR is then ( ) Pavg T d B G t G r λ 2 σn p SNR c = n p (4π) }{{} 3 R 4 = P avgt d G t G r λ 2 σ kt 0 F BL s (4π) 3 R 4 kt 0 F L s = peak P t per pulse 31 / 58

Pulse compression There seems to be two conflicting requirements: High resolution requires short pulse time τ (or rather, high bandwidth) High SNR requires long pulse time τ These requirements can be combined using pulse compression, explained in Chapter 20. The average power form of the RRE remains the same, where SNR pc = P avgt d G t G r λ 2 σ (4π) 3 R 4 kt 0 F L s P avg T d is the energy in one pulse train kt 0 F is the thermal energy in the receiver 32 / 58

Case study: hypothetical radar system SNR Transmitter: 150 kilowatt peak power Frequency: 9.4 GHz Pulse width: 1.2 microseconds PRF: 2 kilohertz Antenna: 2.5 meter diameter circular antenna (an efficiency η = 0.6 is used to determine antenna gain.) Processing dwell time: 18.3 milliseconds Receiver noise figure: 2.5 db Transmit losses: 3.1 db Receive losses: 2.4 db Signal processing losses: 3.2 db Atmospheric losses: 0.16 db/km (one way) Target RCS: 0 dbsm, 10 dbsm (1.0 and 0.1 m 2 ) Target range: 5 to 105 km 34 / 58

Case study, graphical form Different detection ranges for the two different targets. 35 / 58

Search application A solid angle Ω is being scanned for targets at M beam positions with dwell time T d. The total time to scan is then T fs = MT d Ω T d θ 3 φ 3 where θ 3 and φ 3 are the azimuth and elevation 3 db beamwidths. Using θ 3 φ 3 λ 2 /A e and G = 4πA e /λ 2, the average power RRE can be written ( ) ( ) P avg A e R 4 Ω SNR min 4πkT 0 F L s σ where user terms are on the right and system designer terms on the left. This shows that the power-aperture product P avg A e has to be maximized in order to search a big solid angle Ω at small time T fs. 36 / 58 T fs

Track application When tracking one or several targets, important parameters are Tracking precision σ θ 1/ SNR Number of tracked targets N t Updates per second r 37 / 58

RRE for track application The RRE can be rewritten in terms of the tracking parameters as (see derivation in the book, Section 2.16) P avg A 3 ek 2 m λ 4 kt 0 F L s = ( ) ( π 2 rnt R 4 ) ( 2 σ σ 2 θ 1 cos 5 (θ scan ) where k m [1, 2] is a tracking system parameter, and the factor cos 5 (θ scan ) accounts for gain loss and beam broadening when scanning a phased array. This shows the strong dependence on antenna aperture for efficient tracking. With known SNR rather than σ θ, we could also write P avg A 2 e L s F λ 2 = SNR 4πR4 kt 0 PRF σ which also demonstrates a strong dependence on A e. ) 38 / 58

Some trade-offs SNR = P avgt d G t G r λ 2 σ (4π) 3 R 4 kt 0 F L s = P avgt d A et A er σ/λ 2 4πR 4 kt 0 F L s Stealth technology: SNR σ/r 4 shows that σ needs to be reduced significantly in order to affect detection range R. This implies high costs. SNR increases with increased dwell time T d, at the expense of longer measurement times. For fixed A e and σ (antenna and scatterer large compared to wavelength), smaller wavelength increases SNR. 39 / 58

Task of the search mode The task of the search mode is to scan through a certain volume, and detect the presence of targets with no a priori knowledge of their existence. The radar beam is directed at different angles, mechanically or electrically, and measurements are taken at each position. The scan has to be fast enough, so targets do not have too much time to move. 42 / 58

Mechanical vs electrical scanning Mechanical Rotating turret. Typically scans in azimuth. Continuous movement 360 one direction or finite sector back-and-forth. Rotation speed needs to align with dwell time and range delay. Electrical Phased array. Scans quickly in all directions. Beam width changes with angle. Scan loss can be compensated by increasing dwell time at large angles. Search can be combined with track either by tracking-while-scanning (slow update), or search-and-track (interleaving track function, only ESA). 43 / 58

Threshold concept A detection is registered when a signal is registered above a threshold, giving some margin to the noise floor. The signal needs to be considered as a random variable. 45 / 58

Probability Probability of False Alarm: P FA = Probability of Detection: P D = V t V t p i (v) dv p s+i (v) dv The probability density function (PDF) is denoted p, index i for interference and index s+i for signal in the presence of interference. 46 / 58

Noise probability distribution: Rayleigh distribution When measuring both amplitude and phase, v = I + jq, the I and Q signals due to noise are zero mean Gaussian. This implies the amplitude r = I 2 + Q 2 is Rayleigh distributed, that is, p i (r) = r ) σn 2 exp ( r2 2σn 2 where σ 2 n is the mean square voltage, or variance of the noise, called noise power. In the figure below, σ 2 n = 0.04. 47 / 58

Probability of false alarm Using the Rayleigh distribution, the probability of false alarm can be computed explicitly (a truly rare case!): P FA = V t r σ 2 n ) exp ( r2 2σn 2 dr = exp ( V t 2 ) 2σn 2 For a desired P FA, this provides the required threshold: V t = 2σ 2 n ln(1/p FA ) To further reduce the P FA, it is common to make confirmation measurements of a detection. With n confirmations, we get P FA (n) = [P FA (1)] n for a false alarm in all dwells, which quickly reduces the P FA. 48 / 58

Signal + noise PDF: Rician distribution For a non-fluctuating target signal embedded in Gaussian noise, we obtain the Rice distribution: ( ) p s+i (v) = v σn 2 exp v2 + v 2 s+i 2σ 2 n I 0 (vv s+i /σ 2 n) where v s+i is the mean amplitude, and I 0 is the modified Bessel function of the first kind and second order. For v s+i = 0 this is the Rayleigh distribution. The probability of detection is ( ) P D = V t p s+i (v) dv = V t v σ 2 n exp v2 + v 2 s+i 2σ 2 n I 0 (vv s+i /σ 2 n) dv Not surprisingly, this does not have a closed solution, but can be implemented numerically (Marcum s Q-function). 49 / 58

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Receiver operating curves, ROC To investigate the trade-off between P FA, P D, and SNR, a curve of two with varying values of the third can be plotted. Higher level system requirements determine desired P FA and P D which can vary a lot. Typical values could be P D around 50% 90% for a P FA in the order of 10 4 10 6. In the curves above, this requires SNR around 10 db 13 db. 51 / 58

Fluctuating targets: motivation Big, real life targets have very complicated RCS, depending strongly on angle. A statistical description is necessary. 52 / 58

The Swerling models Two different PDF:s (different target characteristics), combined with two different fluctuation rates: dwell-to-dwell or pulse-to-pulse. For fixed P FA and P D, a fluctuating target (SW1-4) requires higher SNR than a non-fluctuating target (SW0). Further details in Chapter 7. 53 / 58

Receiver operation curves For high P D the required SNR for fluctuating targets is significantly higher than for non-fluctuating. For low P D fluctuating targets may require lower SNR than fluctuating, but this is seldom an interesting region. 54 / 58

Closed-form solutions Chapter 3.3.8 presents some closed form solutions for computing probabilities. These can be convenient in specific cases, but please always check carefully the region of applicability of the formulas! 55 / 58

Multiple dwells The single-dwell detection probability can be improved using multiple dwells with only a small penalty in cumulative false alarm: P D (n) = 1 [1 P D (1)] n P FA (n) = np FA (1) Requiring m-of-n detections can further improve both probabilities: 56 / 58

Conclusions The radar range equation estimates the received power or SNR. An overview of losses have been presented. The use of multiple pulses to strengthen SNR has been demonstrated. The RRE can be adapted to various specific applications, like search or track, with parameters specific for the application. Fundamentals of detection theory have been treated, introducing P D and P FA. 58 / 58